While exploring a cave, a spelunker starts at the entrance and moves the following distances: 75.0 m north, 275 m east, 215 m at an angle 30.0° north of east, and 150 m south. Find the resultant displacement from the cave entrance.
break it down to horizontal(x) and vertical(y) components!
y = 75m + 215(sin30)m
x = 275m + 215(cos30)m
perform Pythagorean theorem!
sqrt(x^2+y^2)=?
displacement also has direction, so
tan^-1(y/x)=?
To find the resultant displacement from the cave entrance, we need to find the sum of the individual displacements.
The figure provided indicates that the spelunker moves north, east, at an angle north of east, and then south. We can break down these displacements into their x (east-west) and y (north-south) components.
First, let's determine the x and y components of each displacement:
- The north displacement of 75.0 m has no east-west component (x), only a north-south component (y) with a magnitude of +75.0 m.
- The east displacement of 275 m has only an east-west component (x) with a magnitude of +275 m. Its north-south component (y) is zero since it is purely in the east direction.
- The displacement 215 m at an angle 30.0° north of east can be divided into its x and y components using trigonometry. The x component (east-west) is given by cos(30°) * 215 m, and the y component (north-south) is given by sin(30°) * 215 m.
- x component = cos(30°) * 215 m = 186.68 m (rounded to two decimal places)
- y component = sin(30°) * 215 m = 107.5 m
- The south displacement of 150 m has no east-west component (x), only a north-south component (y) with a magnitude of -150 m.
Now, we can sum up the x and y components to find the resultant displacement:
x-component = 0 m (from the north displacement) + 275 m (from the east displacement) + 186.68 m (from the angled displacement) + 0 m (from the south displacement) = 461.68 m (rounded to two decimal places)
y-component = 75.0 m (from the north displacement) + 0 m (from the east displacement) + 107.5 m (from the angled displacement) - 150 m (from the south displacement) = 32.5 m
Finally, we can determine the magnitude and direction of the resultant displacement using the Pythagorean theorem and trigonometry:
Magnitude = √(x-component^2 + y-component^2) = √(461.68^2 + 32.5^2) ≈ 464.40 m (rounded to two decimal places)
Direction: tan^(-1)(y-component / x-component) = tan^(-1)(32.5 / 461.68) ≈ 3.99° (rounded to two decimal places) north of east.
Therefore, the resultant displacement from the cave entrance is approximately 464.40 m at an angle of 3.99° north of east.